Question

what is the minimum value of ( f(x)=xln(x) )?
a -e
b -1
c ( -\frac{1}{e} )
d 0
e has no minimum value

Explanation:

Step1: Find the derivative of the function

The function is \(f(x)=x\ln(x)\), and its domain is \(x>0\).
Using the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = x\), \(u^\prime=1\), \(v=\ln(x)\), \(v^\prime=\frac{1}{x}\).
So \(f^\prime(x)=\ln(x)+x\times\frac{1}{x}=\ln(x) + 1\).

Step2: Find the critical points

Set \(f^\prime(x)=0\), then \(\ln(x)+1 = 0\).
Solve for \(x\): \(\ln(x)=- 1\), and by the property of logarithms \(x = e^{-1}=\frac{1}{e}\).

Step3: Determine the nature of the critical point

Take the second - derivative. \(f^\prime(x)=\ln(x)+1\), then \(f^{\prime\prime}(x)=\frac{1}{x}\).
When \(x = \frac{1}{e}\), \(f^{\prime\prime}(\frac{1}{e})=e>0\). So \(x=\frac{1}{e}\) is a point of minimum.

Step4: Calculate the minimum value

Substitute \(x = \frac{1}{e}\) into \(f(x)\): \(f(\frac{1}{e})=\frac{1}{e}\ln(\frac{1}{e})\).
Since \(\ln(\frac{1}{e})=-1\), then \(f(\frac{1}{e})=-\frac{1}{e}\).

Answer:

C. \(-\frac{1}{e}\)